What does "provided" mean?

Provided: On condition; by stipulation; with the understanding; if; -- usually followed by that; as, provided that nothing in this act shall prejudice the rights of any person whatever. Provided the deductions are logical, they seem almost indifferent to their truth. G. H. Lewes. Note: This word is strictly a participle, and the word being is understood, the participle provided agreeing with the whole sentence absolute, and being equivalent to this condition being previously stipulated or established.